Projecting the phase-space trajectory of multidimensional non-equilibrium systems onto a discrete set of states: a Projective Dynamics approach

Phase-space trajectories, which are either continuous or possess small discontinuities, can be projected onto a discrete set of states with nearest neighbor coupling. The pointwise projection leads for non- equilibrium system to a non-Markovian process, even if the dynamics of the original system is Markovian. However, using time-averaged transition-rates a Markov process can be obtained, which has the same overall properties as the original dynamics of the system. The projected process defines a new dynamics, which only in the limit $t\rightarrow \infty$ obtains the property on the time-scale of the averaging procedure. We demonstrate the Projective Dynamics method in theory and applications to absorption processes, which in general are not describable through equilibrium or steady-state models. We show the discrete set of states $\{\zeta_k\}$ can be chosen arbitrarily (with slight restrictions) for all systems.